jimbo007 said:hey there,
i'm curious as to why they call it fractional Brownian motion. please don't say its Brownian motion that is fractional
many thanks
Here you go -- from the last hit on the first page of the search that I posted:jimbo007 said:
Fractional Brownian motion (FBM) is a type of stochastic process that is used to model the behavior of random systems. It is a generalization of the traditional Brownian motion, which is a random walk process with independent and identically distributed increments. FBM allows for the correlation of increments, meaning that each step is not completely independent from the previous ones. This makes FBM a more realistic model for many real-world phenomena.
The main difference between FBM and traditional Brownian motion is the correlation of increments. In traditional Brownian motion, each step is completely independent from the previous ones, whereas in FBM, there is a positive correlation between increments. This means that FBM is more likely to exhibit long-range dependence, while traditional Brownian motion does not.
FBM has various applications in different fields, including finance, geology, physics, and signal processing. In finance, FBM is used to model stock prices and financial returns. In geology, it is used to model the spatial distribution of natural resources. In physics, FBM is used to study the behavior of fluids and polymers. In signal processing, it is used to analyze and filter time-series data.
FBM can be simulated using different methods, such as the Davies-Harte algorithm, the Cholesky decomposition method, or the Hosking method. These methods involve generating a sequence of correlated random variables with a specific covariance structure. The simulated FBM can then be used to generate sample paths that exhibit the desired properties, such as long-range dependence and self-similarity.
The Hurst exponent is a parameter that characterizes the behavior of FBM. It is used to measure the degree of long-range dependence in the process. A Hurst exponent of 0.5 corresponds to traditional Brownian motion, while values greater than 0.5 indicate positive correlation and values less than 0.5 indicate negative correlation. The Hurst exponent is a crucial parameter in the simulation and analysis of FBM, and it can also provide insights into the underlying dynamics of the system being modeled.